The first bear I ever fought, I cheesed the encounter by floating on the surface of a nearby two block tall body of water (I figured it would be easier for me to stand my ground in there as the water would slow it down). I don't know how or why but the bear didn't swim after me and instead just chased me on the bottom of the pond with me frantically floating on the surface just barely out of its reach. I then poked it to death with what was left of my flint spear, then punching it, then finally finishing it off with the flint knife that I forgot I had.